Non-Hermitian swallowtail catastrophe revealing transitions across diverse topological singularities

Abstract Exceptional points are a unique feature in non-Hermitian systems, where eigenvalues and their corresponding eigenstates of a Hamiltonian coalesce. A lot of intriguing physical phenomena arise from the topology of exceptional points, such as “bulk Fermi-arcs” and braiding of eigenvalues. Here we report that a more exotic and structurally richer degeneracy morphology, known as the swallowtail catastrophe in singularity theory, can naturally exist in non-Hermitian systems with both parity-time and pseudo-Hermitian symmetries. The swallowtail exhibits the coexistence and intriguing interactions of degeneracy lines of three different types, including an isolated nodal line, a pair of exceptional lines of order three and a non-defective intersection line, with the latter two types lying entirely on the exceptional surface. Surprisingly, these a priori independent types of singularities are stably connected at a single point, i.e. the vertex of the swallowtail, revealing mutual transitions among them. Moreover, we realized such systems in a non-reciprocal circuit and experimentally observed the degeneracy features of the swallowtail. Based on the frame rotation and deformation of eigenstates, we further demonstrated in theory and experiments that the various transitions are topologically protected. Our findings constitute the first demonstration of a swallowtail structure in band dispersions, en route establishing a whole new family of non-Hermitian topological phases of matter. The transitions across diverse singularities pave new avenues for the development of sensing and absorbing devices.